For winding up flat webs more specifically in the form of narrow film webs, it is conventional to use winding shafts which carry spaced winding cores which they hold by expanding in diameter. The flat webs arranged side by side are then wound on the cores to form rolls. In order to mount the winding cores on the winding shafts the individual winding cores are slipped by hand onto them, then aligned on the shafts and tightly engaged by expanding the shafts. This manner of mounting the winding cores on the winding shafts is not only a slow process but also calls for a substantial amount of concentration on the part of the workforce to align the winding cores in relation to each other properly.